Process Modeling, Simulation & Optimization — Winter 2018

Describe in detail the principles of formulation of mathematical models.

Explain the fundamental laws of physics and chemistry with their applications to simple chemical systems.

Consider a batch reactor in which the following first-order consecutive reactions are carried out. A k 1B k 2C Reactant Ais charged into the vessel. Steam is fed into the jacket to bring the reaction mass up to a desired temperature. Then cooling water must be added to the jacket to remove the exothermic heat of reaction and to make the reactor temperature follow the prescribed temperature-time curve. This temperature profile is fed into the temperature controller as a set-point signal. Derive the temperature profiles for the process and metal wall for the batch reactor described above.

Write the various equations of motion for process modeling.

List the various professional simulators and equation solver software.

Consider the vapourizer sketched in the figure. Liquefied petroleum gas (LPG) is fed into a pressurized tank to hold the liquid level in the tank. We will assume that LPG is a pure component: propane. The liquid in the tank is assumed perfectly mixed. Heat is added at a rate Qto hold the desired pressure in the tank by vapourizing the liquid at a rate W (mass per v time). Heat losses and the mass of the tank walls are assumed negligible. Gas is drawn off the top of the tank at a volumetric flow rate F . Fis the forcing v v function or load disturbance. Derive the model equations for the system for steady state model and liquid and vapour dynamics model.

List the structural components of general purpose sequential modular program.

Find the values of x and z (both > 0) that maximize the function: U = -x2 + 10x + xz – z2 + 8z + 21

Aposter is to contain 300 cm2 of printed matter with margins of 6 cm at the top and bottom and 4 cm at each side. Find the overall dimensions that minimize the total area of the poster.

AkB An irreversible, exothermic reaction is carried out in a single perfectly mixed CSTR as shown in figure. The reaction is nth-order in reactant Aand has a heat of reaction 𝜆 (Btu/lbmol of Areacted). Negligible heat losses and constant densities are assumed. To remove the heat of reaction, a cooling jacket surrounds the reactor. Cooling water is added to the jacket at a volumetric flow rate F , and with an inlet J temperature of T . The volume of water in the jacket Vis constant. The mass Jo J of the metal walls is assumed negligible so the thermal inertia of the metal need not be considered. Derive the model equations with the assumption of a perfectly mixed cooling jacket.

State objective functions in terms of the adjustable variable for chemical reactor.

Abox with a square base and open top is to hold 1000 cm3. Find the dimensions that require the least material (assume uniform thickness of material) to construct the box.

Minimize the quadratic function: f(x) = x2 – x using quasi-newton method.

Explain random search and grid search method for unconstrained multivariable optimization.

Discuss feature of basic tearing Algorithm.

Classify the methods to solve unconstrained multivariable problems.

The total annual cost of operating a pump and motor (C) in a particular piece of equipment is a function of the size (horsepower) of the motor (X),4500 C = 500+X + X Find the motor size that minimizes the total annual cost. Use Newton’s method from the starting point of X = 10. Does the solution converge? Solve the o equation analytically and determine actual solution.

Discuss the optimizing recovery of waste heat with suitable figure and equations.

Determine convexity or concavity for the following functions. f(x) = 4x2 +6x x +3x2 +5x2 +x x - 3x - 2x +

Explain the application of optimization in fitting vapour-liquid equilibrium data.

The analysis of labor costs involved in the fabrication of heat exchangers can be used to predict the cost of a new exchanger of the same class. Let the cost be expressed as a linear equation.2 C = 𝛽 + 𝛽 A+𝛽 N012 Where β , β , and β are constants, N = number of tubes, A = shell surface area.012 Estimate the values of the constants β , β and β from the data in following123 table. Labor cost ($) Area (A), m2 120 130 108 110 84 90 80 55 64 50 Number of tubes (N)

Explain black box model.

Minimize f(x) = x4 −x +1 using Newton’s method. Take starting point = 0.64

Solve the following non-linear function with constraints using Lagrange multiplier method. Minimize: f(x, y) = Kx−1y−2, Subject to: g(x,y) = x2 +y2 = a2